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Journal of Engineering Mathematics ISSN 0022-0833Volume 109Number 1 J Eng Math (2018) 109:227-238DOI 10.1007/s10665-017-9948-0

Modelling peeling- and pressure-drivenpropagation of arterial dissection

Lei Wang, Nicholas A. Hill, StevenM. Roper & Xiaoyu Luo

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J Eng Math (2018) 109:227–238https://doi.org/10.1007/s10665-017-9948-0

Modelling peeling- and pressure-driven propagationof arterial dissection

Lei Wang · Nicholas A. Hill ·Steven M. Roper · Xiaoyu Luo

Received: 2 March 2017 / Accepted: 30 November 2017 / Published online: 13 December 2017© The Author(s) 2017. This article is an open access publication

Abstract An arterial dissection is a longitudinal tear in the vessel wall, which can create a false lumen forblood flow and may propagate quickly, leading to death. We employ a computational model for a dissection usingthe extended finite element method with a cohesive traction-separation law for the tear faces. The arterial wallis described by the anisotropic hyperelastic Holzapfel–Gasser–Ogden material model that accounts for collagenfibres and ground matrix, while the evolution of damage is governed by a linear cohesive traction-separation law.We simulate propagation in both peeling and pressure-loading tests. For peeling tests, we consider strips and discscut from the arterial wall. Propagation is found to occur preferentially along the material axes with the greateststiffness, which are determined by the fibre orientation. In the case of pressure-driven propagation, we examine acylindrical model, with an initial tear in the shape of an arc. Long and shallow dissections lead to buckling of theinner wall between the true lumen and the dissection. The various buckling configurations closely match those seenin clinical CT scans. Our results also indicate that a deeper tear is more likely to propagate.

Keywords Arterial dissection · Cohesive traction-separation law · Collagen fibres · HGO constitutive law · Tear ·Wall buckling · XFEM

1 Introduction

Arterial dissection is a cardiovascular disease with a high mortality rate. An arterial dissection begins when a defectarises in the intimal layer of the arterial wall. The defect tends to grow to become a tear in the medial layer; bloodleaks into the tear to create a false lumen. Subject to loading, the tear may propagate along the media, which canlead to death as a result of decreased blood supply to other major organs, damage to the aortic valve, and sometimes

N. A. Hill · S. M. Roper · X. Luo (B)School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UKe-mail: xiaoyu.luo@glasgow.ac.uk

N. A. Hille-mail: nicholas.hilll@glasgow.ac.uk

S. M. Ropere-mail: Steven.Roper@glasgow.ac.uk

L. WangDepartment of Engineering, Durham University, Durham, DH1 3LE, UK

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rupture of the artery [1,2]. In this paper, we develop computational models of arterial dissection to understandthe mechanical issues surrounding the development of an existing tear in the arterial wall. We define two types ofpropagation of a dissection. We do not consider the dynamics of tear propagation; instead, we are concerned withidentifying a critical value of the loading parameter at which tear propagation might be expected to occur. The first,which we identify as ‘peeling-driven’ propagation, has displacement control on the boundaries of sections of thearterial wall and has zero traction on the tear faces. The second, which we identify as ‘pressure-driven’ propagation,has pressure loading on the tear faces and the lumen of the blood vessel.

Peeling-driven propagation describes a standard test for the characterisation of material strength, in which atorn sample is subjected to displacement boundary conditions. Gasser and Holzapfel [3] developed an eXtendedFinite Element Method (XFEM) model for such a configuration. Ferrara and Pandolfi [4] used the conventionalfinite element method to model the same problem. Both groups used the Holzapfel–Gasser–Ogden (HGO) strain-energy function to describe the mechanical response of the anisotropic arterial wall, but different cohesive traction-separation laws were used for the evolution of the tear; it is an exponential isotropic function of separation in [3],but a linear function with fibre-dependent directional preference in [4]. These computational models were usedto model peeling experiments on the human aortic media by Sommer et al. [5]. Similar work was carried out forpeeling experiments on human coronary [6] and carotid [7] arteries.

Pressure-driven propagation is an alternative experimental protocol to study the mechanics underlying arterialdissection. Carson et al. [8] measured the value of pressure to initiate a dissection in strips of porcine upper thoracicaortas, aswell as curves of pressure against the volumeof the false lumen.Taking a slab of tissue from the longitudinaldirection of the aortas with three layers (intima, media and adventitia), and infusing a fluid at a constant flow rateinto the media, they found that as the pressure increased to 77kPa, a bleb formed, and the tear started to propagate,at which point the flow into the dissection was halted. Propagation continued, with the pressure decreasing as thevolume increased before stopping at a pressure of around 25kPa. Tam et al. [9] performed a pressure-driven tearpropagation in the porcine aortic wall assuming the artery natural geometry to be a cylindrical tube. An initial falselumenwas created by injecting saline solution into themedia, a radial–circumferential slit was introduced to connectthe true and false lumens, which ensures the same pressure in both lumens. Each aorta was then submerged in 0.9%saline and pressurised under static conditions. The pressure was increased up to the critical value corresponding tothe onset of propagation. Introducing the initial tear at different radial distances (depths) from the inner radius ofthe aorta, they found that the critical pressure decreased almost linearly with depth.

Pressure-driven propagation is closer to in vivo loading environment of the arterial dissection, as well as tohypertension-induced dissection in vivo [10], than the peeling-driven case. However, there are few modelling stud-ies on pressure-driven dissection propagation. Rajagopal et al. [11] presented a mathematical model describingmechanical factors in the initiation and propagation of a dissection. They postulated that the haemodynamic con-ditions that render the aorta susceptible to the initiation of dissection are elevated maximum systolic and meanaortic blood pressure, whereas the haemodynamic conditions that facilitate propagation of a dissection are ele-vated pulse pressure and heart rate. Although the equations for the nonlinear viscoelasticity of the arterial wall andfor delamination of the individual layers composing the aortic wall were also included, their model has not beenapplied to any examples. In [12], we computed the energy release rate for a mathematical model of pressure-drivendissection propagation in two-dimensional arterial wall strips, and observed that the connective tissue may resultin dissection arrest, i.e. the tear propagation can be stopped after first spread due to the support of the connectivetissues. In other words, diseased connective tissues enhances the risk of dissection. Recently, we investigated theeffect of residual stress on the critical pressure for dissection propagation in [13], which demonstrated that residualstress can elevate the critical pressure and thus lower the risk propagation. Gültekin et al. [14] used a numericalphase-field approach and, after parameter fitting, found good agreement with experiments by Sommer et al. [15]on shear-loading tests of both aneurysmatic and dissected human thoracic aortas that showed that the orthotropicmedia has significantly greater resistance to out-of-plane than to in-plane shear loading. Sommer et al. also foundthat aortas with dissections had on average much less mechanical strength than aneurysmal specimens. Tsamis etal. [16] conducted a major review of published data and concluded that high blood pressure and degradation of theboth elastin and collagen in the wall contribute to the formation of dissections in the thoracic aorta.

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Modelling peeling and pressure-driven propagation of arterial dissection 229

In this paper, we simulate dissection propagation subject to both peeling and pressure.We assume the arterial wallto be an fibre-reinforced hyperelastic incompressible material, and employ the HGO strain-energy function. Tearinitiation and propagation are governed by a linear cohesive traction-separation law. Solutions are computed usingthe XFEM [17] implemented in Abaqus [18]. For illustration, the simulations of peeling-driven tear propagationin strips and discs of arterial wall samples are used to study the direction of tear propagation. Both plane-strainand three-dimensional models are developed. The conclusions from running these models suggest that the teartends to propagate along the direction of fibre orientation. Simulations of pressure-driven tear propagation in thecross-section of two-layer arterial wall are performed to investigate the effects of tear length and depth on thecritical condition and propagation. The results show that the collapse of wall flap (the layer between the true andfalse lumina) for a long and shallow dissection may slow down the tear propagation. These deformed bucklingconfigurations are not uncommon, as demonstrated by a number of published in vivomedical images. We concludethat flap thickness may be an important factor in predicting the evolution of dissections.

The paper is organised as follows. In Sect. 2, the mathematical models are described. Section 3 gives the resultsof the peeling-driven case, and Sect. 4 studies pressure-driven propagation. The conclusions are drawn in the finalsection.

2 The computational model

2.1 Material

We assume the arterial wall to be a hyperelastic incompressible material that can be described by the HGO strain-energy function [19]:

Ψ ≡ Ψm(I1) + Ψ f (I4, I6) = c(I1 − 3) +∑

n=4,6

ψ(In), (1)

where c is the shear modulus of the ground matrix material, ψ(·) is the contribution to the strain-energy densityfrom the fibres,

ψ (In) |(n=4,6) ={

k12k2

(exp

[k2(In − 1)2

] − 1)

In > 1,

0 In � 1,(2)

and k1 and k2 are material parameters. For two fibre families aligned along the referential unit vector directions, A1

and A2, respectively, the invariants In (n = 1, 4, 6) can be written as

I1 = trC, In = C : Mn, n = 4, 6, (3)

whereC = FFT is the right Cauchy–Green tensor, F is the deformation gradient,M4 = A1⊗A1 andM6 = A2⊗A2

are the structure tensors derived from the fibre orientations. The invariants I4 and I6 are the square of the stretchratios in the directions of A1 and A2, respectively. Therefore, I4 and I6 measure the deformation of the fibres.

2.2 Damage

We use a linear cohesive traction-separation law to govern the behaviour of a torn part of the material [13]. Thereare two independent material parameters in the cohesive law, the maximum traction Tc that the material can bearwithout any damage, and the maximum separation or displacement�uc of surfaces while still maintaining cohesiveforces between them.

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230 L. Wang et al.

Damage is activated at any point where the maximum principal stress σ1 exceeds Tc, i.e.

σ1 > Tc. (4)

Note that a compressive stress does not initiate damage.The damage is activated at a material point when the damage criterion (4) is first satisfied. The cohesive traction-

separation law is then used to relate surface tractions to jumps in displacement. Analogously to the stress–strainrelation for elasticity, the cohesive law relates the traction T and separation�u. There are no definitive experimentaldata on the traction-separation law for large arteries subject to dissection, and so, following Ferrara and Pandolfi[4], we have chosen a simple linear traction-separation law,

T ={Tc (1 − �u/�uc) 0 � �u � �uc,

0 �u > �uc.(5)

Generally, the fracture energy,

Gc = Tc �uc/2, (6)

i.e. the total energy per unit area required to separate two surfaces by the critical separation �uc, is specified in thecomputational model instead of �uc.

2.3 Extended finite element method

The initiation and propagation of a tear is implemented using the extended finite element method (XFEM) [3,17].Compared to a conventional FEM, the displacement in XFEM captures discontinuities, such as tears, by enrichingthe displacement field. Consider X, a point in a FE model. Assume there is a discontinuity in the arbitrary domaindiscretised into n-node finite elements. In the XFEM, the approximation for calculation of the displacement for thepoint X within the domain is

uh(X) = uFE + uEnr =n∑

I=1

NI (X)uI +m∑

J=1

NJ (X)H(X)aJ , (7)

where uI is the vector of regular degree of nodal freedom in the FEM, aJ is the vector of the additional m degreesof freedom and H(X) is the discontinuous enrichment function defined for the set of nodes that the discontinuityhas in its domain of influence. For example, to account for a displacement jump across a tear Γ , H(X) may bedefined as

H(X) ={1 if (X − XΓ ) · n � 0,

−1 otherwise,(8)

where XΓ ∈ Γ is the closest point to X on the tear and n is a outward-pointing normal vector of Γ at XΓ . Thefirst term on the RHS of (7) is the conventional finite element approximation, including all nodes, while the secondterm is the enrichment approximation which takes into account the existence of any discontinuities, including onlynodes of elements bisected by the tear.

The main advantage of the XFEM is that the discontinuity at the tear is not explicitly expressed in the finiteelement mesh, but is placed in the interpolation of the solvable function uh . Therefore the mesh generation in theXFEM is independent of the tear. This also means that tear surfaces do not need to coincide with the interfacesbetween elements. The path of propagation of the tear is calculated based on the solution for the displacementfield, the criterion (4) and the cohesive traction-separation law (5). The direction of propagation is determined tobe normal to the direction of the first principal stress σ1. We remark that although XFEM is a well-tested techniqueand well suited to our single tear problem, recent approaches such as eigen-erosion [20], and phase field [21] havealso been developed, and could be more powerful in modelling branching and other complex fractures.

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Modelling peeling and pressure-driven propagation of arterial dissection 231

Fig. 1 The sketch of thestrip used in the peeling test[5], the cross-section(pink-shaded area) of whichis the geometry used here.Displacement boundaryconditions with samemagnitude u but oppositedirections are applied on thetwo arms

O

u

u

r

z

θ

O

Table 1 The materialparameters for the samplesused in the peeling test

c (kPa) k1 (kPa) k2 Tc (kPa) Gc (N/m)

1 1 10 5 0.001

3 Peeling-driven propagation

3.1 Peeling-driven propagation in strips

3.1.1 Geometry and boundary conditions

The peeling test performed in [5] is a representative experimental study to determine the failure properties of anarterial wall. In this test, a strip is cut from the arterial wall and themedial layer is isolated by removing the adventitiaand intima. A tear is introduced at one end of the centre plane which is subject to displacement boundary conditions(Fig. 1). We solve this as a two-dimensional plane-strain problem.

The two families of fibres are distributed in the θ z-plane and at 5◦ to the z-axis [4]. Other parameters used in theconstitutive laws are shown in Table 1. The strip has a width of 4mm and height of 1.2mm, the same geometry asused in [3–5]. The height represents the radial thickness of arterial wall, and the length of initial tear is assumed tobe 0.4mm.

3.1.2 Results

In order to find the preferred direction of the tear propagation, three strips with different orientations are simulatedand the deformed configurations are shown in Fig. 2.

The simulations show that the circumferential strip (Strip 3) is most resistant to the tearing, since the lengthof the undamaged part is the longest. The orientation of collagen fibres is only 5◦ to the z-axis direction (Fig. 2),therefore the stiffness is greatest along the z-axis. Figure 2 shows that the most favourable direction of propagationis along the z-axis. This agrees with the clinical observations of arterial dissection.

3.2 Peeling-driven propagation of discs

3.2.1 Geometry and boundary conditions

We next simulate peeling of a disc. The disc is cut from the same sample of arterial wall as used for selection of strips(Fig. 3). The geometry of this disc is consistent with that of the strip. The radius is 4mm and thickness is 1.2mm.

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ro

ri

3

1

2

θ

z

r Strip 1

ri

ro

Strip 2

ri

ro

Strip 3

ri

ro

Fig. 2 The failure status of three strips: red indicates the torn elements, blue undamaged elements, and green the cohesive zone. Strip1 is completely torn when the displacement is u = 3.28 mm and the other two strips can sustain loads until u = 4 mm. The length ofundamaged section in Strip 3 is greater than that in Strip 2

ro

ri

θ

zr

3 2

1 r

z

θ

3 2

1

Fig. 3 The radius of the disc is the same as the length of strips in Fig. 2. A circular tear (red line) is initialised at the edge of disc, ofdepth 0.1 times the disc radius, as in the simulations for strips. There is also a control case without fibres

An initial tear with depth 0.4mm along the radial direction is introduced. Thus, all three of the two-dimensionalstrips studied early are represented in this model, as shown by the cross-sections at 1, 2 and 3 in Fig. 3. Indeed,this three-dimensional model includes strips with all possible fibre orientations.

The boundary conditions are consistent with those in the simulation of peeling strips. The centres of top andbottom circles are fixed to avoid the rigid body motion. The displacements of ± 4 mm in r -direction are applied,respectively, on the two edge surfaces (as shown by the pink areas in Fig. 3).

3.2.2 Results

The disc peeling tests are simulated both with and without fibres. As shown in the Fig. 4, the undamaged sectionof the fibre-free disc remains circular, while the undamaged part of the fibrous disc becomes elliptical. The shortaxis of the ellipse is in the z-direction, showing that the tear tends to propagate in the mean fibre direction. Thisis because (i) the the fibres resist deformation, so the strain and hence the stress is the greatest in the directionperpendicular to the mean fibre axis, and (ii) in the XFEM the tear propagates normal to the direction of the firstprincipal stress (Sect. 2.3).

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Modelling peeling and pressure-driven propagation of arterial dissection 233

(a) (b)

(c) (d)

Without fibres — side view With fibres — side view

Without fibres — top view With fibres — top view

Fig. 4 Peeling test on a disc. The grey regions are the undeformed configurations of the disc in Fig. 3, while the superimposed colouredones show the deformed (peeled apart) configurations. The faces of the tear are red, and the deformed top and bottom surfaces are blue.In c, the undamaged central (grey) region of the fibre-free disc remains circular after peeling, while the undamaged central region ofthe fibrous disc d becomes elliptical with its major axis aligned along the mean fibre direction

4 Pressure-driven propagation

4.1 Geometry and boundary conditions

The length and depth of the tear are both important factors in diagnosing the aortic dissection. In clinical practice,these data are generally extracted from the images, for example, from CT scans of the cross-section of the arterialwall. Here we study these effects on the propagation of arterial dissection through numerical simulations.

Similar to clinical CT images, we consider the propagation of arterial dissection in a cross-section of arterialwall, as shown in Fig. 5. This is a two-dimensional plane-strain problem with the simplifying assumption that thereis no variation in the axial direction. This example captures the planar propagation of the arterial dissection whichcan, however, in vivo assume a spiral path in the out-of-plane direction, e.g. from the aortic root to the aortic arch.We quantify the length of the initial tear using the central angle η. The initial tear is assumed to be located in themedia and its depth from the inner surface is scaled on the thickness of the media, so that d = 0 places the initialtear on the inner surface and d = 1 means it is on the interface between media and adventitia. We shall simulatepressure-driven tear propagation for different values of 0◦ < η < 360◦ and 0 < d < 1.

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234 L. Wang et al.

2βa

2βm Media

Adven

titia

Dm

D

η 13

2

4

Fig. 5 The cross-section of the two-layer model of the rabbit carotid arterial wall is considered as a plane-strain problem. Thepropagations of the initial tear (black arc) in the media subject to pressure (blue arrows) are simulated. The length of the tear ischaracterised by the central angel η. The depth of tear D is normalised by the thickness of the media Dm, so the dimensionless depthis d = D/Dm

Table 2 The material parameters in the HGO stressrain-energy function for the two-layer model of a rabbit carotid artery [19]

c (kPa) k1 (kPa) k2 β (◦) Tc (kPa) Gc (N/m)

Media 1.5 2.3632 0.8393 29 3.0 0.001

Adventitia 0.15 0.5620 0.7112 62 0.3 0.0001

Boundary conditions are applied to prevent rigid body motion: radii 1 and 3 (Fig. 5) are fixed in the y-directionso that only horizontal movement is allowed, while radii 2 and 4 are fixed in the x-direction so that only have thevertical movement is permitted.

We assume the pressure to be the same in both true and false lumina. As the pressure is gradually increased, weestimate the value of pressure when the dissection just starts to propagate at the tear tip, i.e. the critical pressure pc.

The two-layer arterial wall model uses parameters appropriate to a rabbit carotid artery [19]. Each layer ismodelled using the HGO strain-energy function (1) but with different material parameters, as shown in Table 2.The geometric parameters are inner radius Ri = 0.74 mm, thickness of media Dm = 0.26 mm, and thickness ofadventitia Da = 0.12 mm.

4.2 Simulations and results

4.2.1 Values of the tear depth d and length η

For convenience in the XFEM simulation, the values of d and η chosen for simulations are such that the initial tearis always located within an element, and not along the element boundary. In practice, the best place to specify aninitial tear is the bisector of an element. We choose the value of d to be dk = (k + 0.5)�d, k = 0, . . . , n, and�d = 1.0/(n + 1). From a set of simulations with n = 10, we can estimate the critical pressure pc for each valueof d. In the following, the critical pressure pc is non-dimensionlised by c, the shear modulus of the media.

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Modelling peeling and pressure-driven propagation of arterial dissection 235

Fig. 6 The dimensionlesscritical pressure pc isplotted against the radialdepth d for different valuesof η. Note for η ≥ 100◦, theinner wall has a tendency tobuckle, and the hollowversions of the symbols inthe key indicate the cases inwhich the buckling occurs

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

d

pc

5◦

20◦

40◦

60◦

100◦

160◦

300◦

(a) (b) (c) (d)d = 0.23 d = 0.59 d = 0.77 d = 0.95

Fig. 7 The representative shapes of deformed arterial wall with a short tear of η = 20◦ for different radial depths d, subject to thecritical pressure. The direction of initial propagation changes gradually from the radial (a) to the circumferential (d) as d increases

4.2.2 Effects of d and η on the critical pressure pc

The variation of the critical pressure pc against the radial depth d are plotted in Fig. 6, for different values of η. It isinteresting to see that when the tear is very short, η = 5◦, pc increases monotonically with d. When the dissectionis very long, i.e. η ≥ 60◦, pc decreases with d monotonically. However, when η is between 20◦ to 40◦, the changeis no longer monotonic. This may be related to the change in direction of tear propagation, as shown in Fig. 7.

Note that for a longer tear, e.g. η ≥ 100◦, buckling of the inner wall (the material section between the lumenand tear) occurs, as shown by the hollow symbols in Fig. 6. This buckling phenomenon has also been observed in[13] for a tear located halfway through the media. Here, we find that buckling of the inner wall only occurs for tearswith depth d < 0.7, and deeper tears are less likely to induce wall buckling.

Figure 7 shows the failure patterns of the deformed arterial wall with a tear η = 20◦ subject to the criticalpressure for different values of d. When d = 0.23 in Fig. 7a, the tear propagates radially, but when d = 0.95 inFig. 7d it propagates circumferentially. When d = 0.59 and 0.77, the direction of propagation is between the radialand the circumferential directions. Hence, the tear propagation undergoes a direction change, from the radial to thecircumferential direction, as d increases. Similar trends are observed in simulations for η = 10◦, 15◦ and 25◦.

Our results for η > 60◦ show that pc decreases almost linearly with the tear depth d. This agrees with theexperimental observation by Tam et al. [9]. In that study, initial tears of different radial depths were introducedinside of a porcine aortic wall, which was then inflated under an increasing pressure. At the instantaneous timeof tear propagation, the pressure was regarded as the critical pressure. In [9], this critical pressure is called the“propagation pressure”. The relationship showing that the critical pressure is a linearly decreasing function of thedepth as shown in Fig. 2 of [9].

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TL TLFL TLFL TL

(a) (b) (c) (d)d = 0.14, p = 0.26 kPa d = 0.32, pc = 0.49 kPa d = 0.68, pc = 0.34 kPa d = 0.95, pc = 0.2 kPa

Fig. 8 The representative shapes of deformed arterial wall with an initial tear η = 160◦ for different radial depths d. The areas of thetrue lumen (TL) and the false lumen (FL) are marked. The buckling of the inner wall, the material between TL and FL, occurs, as shownin (a) and (b), which are similar to the CT scans of aortic dissection in patients in Fig. 9a and b, respectively

Fig. 9 CT scans with buckling of the inner wall in type A aortic dissection: a Fig. 9 in [22], b Fig. 2 in [10], and c Fig. 2 in [23]

4.2.3 Effects of d and η on the deformed shapes

For a long tear, say, η = 160◦, the inner wall starts to buckle when d is small. This is shown in Fig. 8a–c. Similarbuckling patterns of the inner wall have been observed in CT scans of acute aortic dissections. For example, thebuckling mode of the inner wall in Fig. 8a is similar to that observed in the CT scan in Fig. 9a, and the bucklingmode in Fig. 8b is similar to the CT scan of the ascending aorta in Fig. 9b. Although our model is simplified, it isencouraging that the model can reproduce qualitatively similar patterns observed in clinics.

Comparison of the deformed shapes of the arterial wall with different tear depth d shows that the radial depthof a tear plays an important role in wall deformation, which in turn determines the likelihood of tear propagation.The results in Fig. 8a–d show that the buckling mode is the highest in (a), and this tear does not propagate. Thebuckling mode decreases as d increases from (a) to (c), with no buckling in (d), when the tear is deepest.

In addition, we plot the deformed shapes of the arterial wall with a longer tear η = 340◦ for different values ofd in Fig. 10. In these results, we also observe that the buckling mode of the inner wall with d = 0.23 in Fig. 10c issimilar to that in the CT scan of the aortic dissection in Fig. 9c [23]. For a tear with d = 0.32 and 0.50 (Fig. 10d,e), the collapse of the true lumen occurs as a result of buckling of the inner wall.

5 Conclusions

We have developed computational models to understand arterial dissection. In this paper, the stable displacement-peeling-driven and unstable pressure-driven tear propagation are simulated. The simulations for the peeling testshow the tear prefers to propagate along the direction of material axial with the maximum stiffness, which isdetermined by the fibre orientation.

We also investigate the effects of depth and length on the critical condition and tear propagation under pressure fora two-layer plane-strain arterial wall model. We found that the critical propagation pressure is greater for deeper andshorter tears. Our model can also reproduce a range of inner wall buckling patterns akin to the clinical observationsusing different tear lengths and depths, despite the simplifying assumption of plane-strain that neglects variationand propagation in the axial direction. Our results suggest that a long and shallow tear is more likely to induce the

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Modelling peeling and pressure-driven propagation of arterial dissection 237

TLFL

TLFL

(a) (b) (c)

(d) (e) (f)

d = 0.05, p = 0.039 kPa d = 0.14, p = 0.083 kPa d = 0.23, p = 0.24 kPa

d = 0.32, p = 0.32 kPa d = 0.50, pc = 0.36 kPa = 0.86, pc = 0.17 kPa

Fig. 10 The representative deformed shapes of arterial wall with a long tear η = 340◦ for different radial depth d. When the depth issmall (a–d), the propagation does not occur. Instead, buckling of the inner wall occurs, and the true lumen is collapsed in (d) and (e).The shape of the buckled inner wall in (c) is similar to that in the CT scan of the aortic dissection for a patient in Fig. 9c

inner wall buckling. We hypothesise that a buckled state redistributes the energy and stresses within the wall, andthis may work to slow down the tear propagation. This is supported by the observation that a deep tear withoutbuckling tends to propagate with a lower critical pressure.

Acknowledgements We are grateful for the support provided by the UK Engineering and Physical Sciences Research Council (GrantNo. EP/N014642/1). LW was supported by a China Scholarship Council Studentship and the Fee Waiver Programme of the Universityof Glasgow. XYL wishes to acknowledge funding from the Leverhulme Trust Fellowship (RF-2015-510) and the NSF China (No.11471261). We also thank Ashwani Goel, a Technical Specialist from SIMULIA Central, for his kind help with using Abaqus.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided yougive appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changeswere made.

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